Title

Big Ramsey degrees for finitely constrained binary free amalgamation classes

Abstract

Fix a countable first-order structure K. We say that a finite substructure A has big Ramsey degree k in K if k is the least natural number satisfying the following property: for any coloring of the embeddings of A in K into finitely many colors, there is some embedding \phi of K into itself so that the set of embeddings {\phi\circ f: f an embedding of A into K} is colored in at most k colors. When K is a countably infinite set, we recover the classical infinite Ramsey theorem. When K is the rational linear order, work of D. Devlin characterizes the big Ramsey degree of the n-element linear order precisely. We say that a structure has finite big Ramsey degrees if every finite substructure has finite big Ramsey degree in the structure. We say that a Fraisse class has finite big Ramsey degrees if the Fraisse limit has finite big Ramsey degrees. In the past few years, Dobrinen has shown that the class of finite triangle free graphs has finite big Ramsey degrees; this was the first known example where the class of structures has a constraint, i.e. that there are no triangles. Recenty, Dobrinen has also shown this for the class of k-clique-free finite graphs. In this talk, we will discuss a simplification of the proof which generalized to any finitely constrained binary free amalgamation class.

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